Rosenbrock-Type Methods

Abstract

... is discussed in this section. Among the methods which already give satisfactory results for stiff equations, Rosenbrock methods are the easiest to program. We shall describe their theory in this section, which will lead us to our first “stiff” code. Rosenbrock methods belong to a large class of methods which try to avoid nonlinear systems and replace them by a sequence of linear systems. We therefore call these methods linearly implicit Runge-Kutta methods. In the literature such methods are often called “semi-implicit” (or was it “semi-explicit”?), or “generalized” or “modified” or “adaptive” or “additive” Runge-Kutta methods.

When the functions φ are non-linear, implicit equations can in general be solved only by iteration. This is a severe drawback, as it adds to the problem of stability, that of convergence of the iterative process. An alternative, which avoids this difficulty, is ...

(H.H. Rosenbrock 1962/63)